Check these threads out, its not out yet. but these explain it will be before launch.
https://www.pantheonmmo.com/content/forums/topic/3621/early-name-reservation/view/post_id/56299
https://www.pantheonmmo.com/content/forums/topic/3602/early-name-reservation
Keep in mind that the linked threads are quite old and may, or may not, have anything to do with what VR is currently planning.
Speculation in more recent threads tends to assume that name reservation will come shortly before release for the excellent reasons that by that time they will have finalized how names work and will have the servers set up so they can *remember* the names we have taken. The former is obviously necessary - they wouldn't want us to reserve first and last names a year ahead of release and foreclose the option of switching to just first names. Or vice versa. And while they seem to be leaning towards first and last names this is not set in stone yet. The latter is not necessary they can maintain a separate data base for the reserved names and transfer them to the servers later. But while not necessary it is logical to wait.
Speculation also considers it very possible although not certain that name reservation before release will work the same as name reservation after release - using the character creation screen and possibly involving actual creation of a character to go with the name. Presumably if they do this we can race to create characters with the names we want then delete them after release to spend more time on getting them just right.
Getting one name unique to all servers rather than unique names per server is what they have indicated they are leaning towards but this is not yet certain. It seems likely if we have unique first and last name combinations rather than just first names.
Whether we will all queue for names at the same time or whether it will be in stages has not been discussed by VR to my knowledge.
Whether there is a limitation on names (perhaps one per person perhaps the number of character slots per server perhaps the number of character slots total if this is greater) has not been discussed by VR to my knowledge.
I follow the forums reasonably closely but I do not follow the streams - bad hearing. I may have missed quite a bit and corrections are welcome.
When I pledged Alpha/Beta I got to reserve my full name, including "family" name.